Programmable optical processing device employing stacked controllable phase shifting elements in fractional fourier planes

ABSTRACT

A monolithic or hybrid integrated optical processor or optical processing system having a plurality of phase shifting array elements, each controlled by respective control signals, and arranged so that each phase shifting array element lies in a different fractional Fourier transform plane. In various embodiments, at least a portion of the resulting system is implemented in a stack of element materials. In one embodiment, a segment of graded index material lies between consecutive phase shifting array elements. Other features include obtaining images from an electronically-controllable image source, using an image sensor to change the processed image into an electrical output, and using at least one of the phase shifting array element to introduce a phase shift.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.11/929,259, filed on Oct. 30, 2007, which is a continuation of U.S.application Ser. No. 11/294,685, filed Dec. 5, 2005, which is acontinuation of U.S. application Ser. No. 10/656,342, filed Sep. 4,2003, now U.S. Pat. No. 6,972,905, which is a divisional of U.S.application Ser. No. 09/512,781 entitled “IMAGE PROCESSING UTILIZINGNON-POSITIVE-DEFINITE TRANSFER FUNCTIONS VIA FRACTIONAL FOURIERTRANSFORM” filed Feb. 25, 2000, which claims benefit of priority fromUS. provisional applications Ser. Nos. 60/121,680 and 60/121,958, eachfiled on Feb. 25, 1999. The provisional applications are incorporated intheir entirety herein.

BACKGROUND OF THE INVENTION

1. Field of Invention

This invention relates to optical signal processing, and moreparticularly to the use of fractional Fourier transform properties oflenses with traditional non-phase-shifting optical elements withintraditional Fourier optical signal processing environments to realize,or closely approximate, arbitrary non-positive-definite transferfunctions. The system and method herein can be applied to conventionallens-based optical image processing systems as well as to systems withother types of elements obeying Fractional Fourier optical models and aswell to widely ranging environments such as integrated optics, opticalcomputing systems, particle beam systems, radiation accelerators, andastronomical observation methods.

2. Discussion of the Related Art

A number of references are cited herein; these are provided in anumbered list at the end of the Detailed Description. These referencesare cited as needed through the text by reference number(s) enclosed insquare brackets. Further, the cited disclosure contained withinreference [1-19] is hereby incorporated by reference.

The Fourier transforming properties of simple lenses and related opticalelements is well known and heavily used in a branch of engineering knownas “Fourier Optics” [1, 2]. Classical Fourier Optics [1, 2, 3, 4] allowsfor flexible signal processing of images by (1) using lenses or othermeans to take a first two-dimensional Fourier transform of an opticalwavefront, thus creating at a particular spatial location a “Fourierplane” wherein the amplitude distribution of an original two-dimensionaloptical image becomes the two-dimensional Fourier transform of itself,(2) using a translucent plate or similar means in this location tointroduce an optical transfer function operation on the opticalwavefront, and (3) using lenses or other means to take a second Fouriertransform which, within possible scaling and orientation differences,amounts to the convolution of the impulse response corresponding to theoptical transfer function with the original image. In this way imagescan be relatively easily and inexpensively lowpass-filtered (detailssoftened) and highpass-filtered (details enhanced) as well as multitudeof other possibilities. These multitudes of possibilities have, due toproperties of materials and fabrication limitations in transcendingthem, been limited to transfer functions that mathematically are“positive-definite;” that is, those which affect only amplitude and donot introduce varying phase relationships.

The Fractional Fourier transform has been independently developedseveral times over the years [5, 7, 8, 9, 10, 14, 15] and is related toseveral other mathematical objects such as the Bargmann transform [8]and the Hermite semigroup [13]. As shown in [5], the most general formof optical properties of lenses and other related elements [1, 2, 3] canbe transformed into a Fractional Fourier transform representation. Thisfact, too, has been apparently independently rediscovered some yearslater and worked on steadily ever since (see for example [6]) expandingthe number of optical elements and situations covered, It is importantto remark, however, that the lens modeling approach in the later longongoing series of papers view the multiplicative-constant phase term inthe true farm of the Fractional Fourier transform as a problem orannoyance and usually omit it from consideration; this is odd as, forexample, it is relatively simple to take the expression for lenses from[2] and repeat the development in [5] based on the simplified expressionin [1] and exactly account for this multiplicative-constant phase term.

SUMMARY OF THE INVENTION

A principal aspect of this invention is the use of Fractional Fouriertransform properties of lenses or other optical elements or environmentsto introduce one or mare positive-definite optical transfer functions atvarious locations outside the Fourier plane so as to realize, or closelyapproximate, arbitrary non-positive-definite transfer functions.Specifically this aspect of the invention encompasses an optical systemfor realizing optical filtering through the use of at least one opticalelement outside the Fourier transform plane. By choice of the number ofsuch elements, position of such elements, and the actualpositive-definite transfer function used for each element, arbitrarynon-positive-definite transfer functions can be approximated by theentire system, and designs can be straightforwardly obtained by methodsof approximation. However, there are several additional aspects to theinvention relating to implementing, expanding, or utilizing thisunderlying aspect.

An intimately related aspect of the invention relates to realizations ofthe optical filtering effects of non-positive-definite optical transferfunctions through the use of at least one positive-definite opticalelement outside the Fourier transform plane. The invention thus includescases where only positive-definite optical elements are used to realizenon-positive-definite optical transfer functions.

An additional aspect of the invention pertains to embodiment designswhich can be straightforwardly obtained by methods of mathematicalfunction approximation. An exemplary approximation method provided forin the invention leverages Hermite function expansions of the desiredtransfer function, This is advantageous in simplifying the approximationproblem as the orthogonal Hermite functions diagonalize the Fouriertransform and Fractional Fourier transform; the result is two-fold:

-   -   throughout the entire optical system the amplitude and phase        affairs of each Hermite function are completely independent of        those of the other Hermite functions    -   the Hermite function expansion of a desired transfer function        will naturally have coefficients that eventually tend to zero,        meaning that to obtain an arbitrary degree of approximation only        a manageable number of Hermite functions need be handled        explicitly.

Another aspect of the invention involves the determination of theposition and/or amplitude distribution of a positive-definite opticalelement through use of the fractional Fourier transform.

Another aspect of the invention involves the determination of theposition and/or amplitude distribution of a positive-definite opticalelement through use of Hermite function expansions, with or withoutapproximations.

Another aspect of the invention relates to the use of at least oneprogrammable light modulator as an optical element.

Another aspect of the invention relates to use of the invention toperform general optical computing functions, explicitly including thoseinvolving complex arithmetic.

Another aspect of the invention provides for realization of embodimentsutilizing integrated optics, including monolithic or hybrid fabricationinvolving photolithography and/or beam-controlled fabrication methods.

Another aspect of the invention provides for multiple channelconfigurations which can be used, for example, in visual-color imageprocessing or wide-spectrum image processing.

Another aspect of the invention provides for replacing one or motenon-positive-definite filter elements in the principal and relatedaspects of the invention with one or more controllable optical phaseshift elements, most generally in array form.

Another aspect of the invention uses the above variation (employing oneor more controllable optical phase shift elements) to synthesizecontrollable lens and lens system functions such as controllable zoomand focus.

[Another aspect of the invention provides for adapting the entiremathematical, method, and apparatus framework, so far built around thefractional Fourier transform operation and associated Hermite basisfunctions, to related mathematical transform operations associated withdifferent basis functions that would arrive from non-quadraticgraded-index media andor environments.

The system and method herein can be applied to conventional lens-basedoptical image processing systems as well as to systems with other typesof elements obeying Fractional Fourier optical models and as well towidely ranging environments such as integrated optics, optical computingsystems, particle beam systems, radiation accelerators, and astrologicalobservation methods.

The incorporation of the method of the invention with classical,contemporary, and future methods of Fourier optics has significantsynergistic value in attaining simple low-cost realizations of opticallinear signal processing with non-positive definite transfer functions.Conventional methods for the creation of means for introducingpositive-definite (amplitude variation without phase variation) opticaltransfer functions can be readily used in fabrication and conventionalapproximation methods can be used in transfer function design. Throughuse of Hermite function expansions, as used in one embodiment of themethod, interaction of terms is minimized and only a manageable numberof terms need be handled explicitly.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other aspects, features and advantages of the presentinvention will become more apparent upon consideration of the followingdescription of preferred embodiments taken in conjunction with theaccompanying drawing figures, wherein:

FIG. 1 shows a general arrangement involving an image source, lens orlens system or other equivalent, and an image observation entity,capable of classical geometric optics, classical Fourier optics, andfractional Fourier transform optics;

FIG. 2 shows a Classical Fourier Optics image processing arrangementwith an optical transfer function element introduced in the Fourierplane, using two lenses to realize the Fourier plane;

FIG. 3 shows optics device 100 providing the exemplary introduction oftwo additional optical transfer function elements introduced outside theFourier plane;

FIG. 4 shows optics device 100 of FIG. 3 implemented as a monolithicoptics device; and

FIG. 5 shows optics device 100 of FIG. 3 integrated with an opticssystem.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The invention is concerned with processing presented image data, eitherrecorded or real-time provided by an exogenous system, means, or method.This image data can be presented by means of an electronic display (suchas an LCD panel, CRT, LED array, or other technologies), films, slides,illuminated photographs, or the output of some exogenous system such asan optical computer, integrated optics device, etc. The presented imagedata will herein be referred to as the image source. The invention isalso concerned with image data then produced by the invention which ispresented to a person, sensor (such as a CCD image sensor,photo-transistor array, etc.), or some exogenous system such as anoptical computer, integrated optics device, etc. The latter imagepresentation receiving entity will herein be referred to as a observer,image observation entity, or observation entity.

FIG. 1 shows a general arrangement involving an image source 101, lensor lens system or other equivalent 102, and an image observation entity103, capable of classical geometric optics, classical Fourier optics,and fractional Fourier transform optics. The class of optics (geometric,Fourier, or fractional Fourier) is determined by the following:

the separation distances 111 and 112

-   -   the “focal length” parameter “f” of the lens or lens system or        other equivalent 102.    -   the type of image source (lit object, projection screen, etc.)        in so far as whether a plane or spherical wave is emitted.

As is well known, the cases where the source image is a lit object andwhere the distances 111, which shall be called “a”, and 112, which shallbe called “b,” fall into the “lens-law relationship” determined by thefocal length f:

$\begin{matrix}{\frac{1}{f} = {\frac{1}{a} + \frac{1}{\; b}}} & (1)\end{matrix}$

gives the geometric optics case. In this case the observed image 103 isa vertically and horizontally inverted version of the original imagefrom the source 101 scaled in size by a magnification factor “m” givenby:

$\begin{matrix}{m = \frac{b}{a}} & (2)\end{matrix}$

The Fourier transforming properties of simple lenses and related opticalelements is also well known and heavily used in a branch of engineeringknown as “Fourier Optics” [2,3]. Classical Fourier Optics [2, 3, 4, 5]involves the use of a lens, lens-systems, or other means to take a firsttwo-dimensional Fourier transform of an optical wavefront, thus creatingat a particular spatial location a “Fourier plane” wherein the amplitudedistribution of an original two-dimensional optical image becomes thetwo-dimensional Fourier transform of itself. In the arrangement of FIG.1 with a lit object serving as the source image 101, the Fourier opticscase is obtained when a=b=f. In this way, classical Fourier Optics [2,3, 4, 5] allows for easy, inexpensive, flexible signal processing ofimages.

FIG. 2 shows a Classical Fourier Optics image processing arrangementwith an optical transfer function element 103 introduced in the Fourierplane 104, using two lenses 102, 105 to realize the Fourier plane 104.The optical transfer function element 103 is typically implemented via atranslucent plate or similar means in this location to introduce anoptical transfer function operation on the optical wavefront. The imagesource 101 and a first lens 102 are separated by a distance 111 which,based on the focal length of the lens, creates a Fourier plane 104 at adistance 112 on the opposite side of the first lens 102. The imagesource 102 may be natural image (as with a camera), produced by anoptoelectric transducer, or some other type of image source element. Asecond lens 105 is positioned a distance 113 from the Fourier plane onthe plane's opposite side. The distance 113 is selected, based on thefocal length of the second lens 105, together with distance 114 so thatan observation element 106 receives a Fourier transform of the imageemanating from the Fourier plane 104. The observation element 106 may benatural (such as a viewfinder, display surface, projection screen, orother means), optoelectric (as in a phototransistor or CCD array orother means), or some other type of observing element.

As an aside, note that if the translucent plate is perfectly clear so asto effectively not be present, the result is the composition of twoFourier transform operations; since an inverse Fourier transform is aFourier transform with its transform variable replaced with itsnegative, the composition of two Fourier transforms amounts to reversingthe image coordinates, i.e., flipping the image up for down and left forright, the result expected for a compound lens. Further, the distances111, 112, 113, 114 to give these effects are not unique; alternatedistance selections result in variations in image size, which amounts tovarying the magnification power of a compound lens and hence the imagesize.

Within then possible scaling and flipped-image orientation differences,the above arrangements amount to the convolution of the impulse responsecorresponding to the optical transfer function with the original image.The same results can be obtained in other optical arrangements and withalternate types of elements, either of which may obey the samemathematical relationships. In this way images can be relatively easilyand inexpensively lowpass-filtered (details softened) andhighpass-filtered (details enhanced) as well as multitude of otherpossibilities.

These multitudes of possibilities have, due to properties of materialsand fabrication limitations in transcending them in the construction ofthe transform element 105, been limited to transfer functions thatmathematically are “positive-definite,” i.e. those which affect onlyamplitude and do not introduce varying phase relationships.

The apparatus and method of this invention utilizes Fractional Fouriertransform [5, 7, 8, 9] properties of lenses [5, 6] or other means tointroduce one or more positive-definite optical transfer functions atvarious locations outside the Fourier plane to realize, or closelyapproximate, arbitrary non-positive-definite transfer functions. TheFractional Fourier transform properties of lenses cause complex butpredictable phase variations to be introduced by each such locatedconventional positive-definite (i.e., amplitude variation without phasevariation) optical linear transfer function elements. By choice of thenumber of such elements, position of such elements, and the actualpositive-definite transfer function used for each element, arbitrarynon-positive-definite transfer functions can be approximated by theentire system. This is now explained in more detail below.

As described in [5], for cases where a, b, and f do not satisfy the lenslaw of the Fourier optics condition above, the amplitude distribution ofthe source image 101 as observed at the observation entity 103experiences in general the action of a non-integer power of the Fouriertransform operator. As described in [5], this power, which shall becalled a, varies between 0 and 2 and is determined by an Arccosinefunction dependent on the lens focal length and the distances betweenthe lens, image source, and image observer, specifically as

$\begin{matrix}{\alpha = {\frac{2}{\pi}{arc}\; {\cos \left\lbrack {{{sgn}\left( {f - a} \right)}\frac{\sqrt{\left( {f - a} \right)\left( {f - b} \right)}}{f}} \right\rbrack}}} & (3)\end{matrix}$

for cases where (f-a) and (f-b) share the same sign, There are othercases which can be solved for from the more primitive equations in [5](at the bottom of pages ThE4-3 and ThE4-I). Note simple substitutionsshow that the lens law relationship among a, b, and f indeed give apower of 2 and that the Fourier optics condition of a=b=f give a powerof 1, as required.

Without loss of generality to other implementations, FIG. 3 shows theexemplary realization of the invention introducing additional (oralternate) optical transfer function elements, here exemplified as 103 aand 103 b, introduced at additional (or alternate) locations, hereexemplified as 104 a and 104 b, in a region outside the Fourier plane,here exemplified with region 113. It is understood that, depending onthe desired transfer function to be realized with what degree ofaccuracy, arbitrary numbers of optical transfer function elements suchas 103 a and 103 b could be located in any one or more of the regions111, 112, 113, 114, and in fact the system may include or not include anelement 103 in the Fourier plane 104. Designs can be straightforwardlyobtained by methods of approximation [11, 12].

One embodiment of the approximation method leverages Hermite function[16] expansions [17, and more recently, 181 of the desired transferfunction. This is advantageous in simplifying the approximation problemas the orthogonal Hermite functions diagonalize the Fourier transform[17] and Fractional Fourier transform [5, 9]; the result is two-fold:

-   -   throughout the entire optical system the amplitude and phase        affairs of each Hemite function are completely independent of        those of the other Hermite functions    -   the Hermite function expansion of a desired transfer function        will naturally have coefficients that eventually tend to zero,        meaning that to obtain an arbitrary degree of approximation only        a manageable number of Hemite functions need be handled        explicitly.

Because of these properties it is simplest to explain the properties ofthe Fractional Fourier transform on the image in terms of the Hermitefunction representation. It is understood that other methods forrepresenting properties of the Fractional Fourier transform on an imageas well as for approximation methodology are possible.

A bounded (non-infinite) function k(x) can be represented as an infinitesum of Hennite functions {h_(n),(x)} as:

$\begin{matrix}{{k(x)} = {\sum\limits_{n = 0}^{\infty}{a_{n}{h_{n}(x)}}}} & (4)\end{matrix}$

Since the function is bounded the coefficients {se}ve ntually becomesmall and converge to zero. An image is a two dimensional entity, as isthe amplitude variation of a translucent plate; in either case thefunction can be represented in a two-dimensional expansion:

$\begin{matrix}{{k\left( {x_{1},x_{2}} \right)} = {\sum\limits_{m = 0}^{\infty}{\sum\limits_{n = 0}^{\infty}{a_{n,m}{h_{n}\left( x_{1} \right)}{h_{m}\left( x_{2} \right)}}}}} & (5)\end{matrix}$

For simplicity, consider the one-dimensional case. The Fourier transformaction on Hermite expansion of the function k(x) with seriescoefficients {a_(n)} is given by [16]:

$\begin{matrix}{{F\left\lbrack {k(x)} \right\rbrack} = {\sum\limits_{n = 0}^{\infty}{\left( {- } \right)^{n}a_{n}{h_{n}(x)}}}} & (6)\end{matrix}$

Because of the diagonal eigenfunction structure, fractional powers ofthe Fourier transform operator can be obtained by taking the fractionalpower of each eigenfunction coefficient. The eigenfunction coefficientshere are (—i)^(n). Complex branching artifact ambiguities that arisefrom taking the roots of complex numbers can be avoided through writing−i as

e^(−iπ/2)  (7)

Thus, for a given power α, the Fractional Fourier transform of theHermite expansion of the function k(x) with series coefficients {a_(n)}sgiven by [5]:

$\begin{matrix}{{F^{\alpha}\left\lbrack {k(x)} \right\rbrack} = {\sum\limits_{n = 0}^{\infty}{^{\; n\; {{\pi\alpha}/2}}a_{n}{h_{n}(x)}}}} & (8)\end{matrix}$

Thus as the power a varies with separation distance (via the Arccosinerelationship depending on separation distance), the phase angle of thenth coefficient of the Hermite expansion varies according to therelationship shown above. When α=I the result is the traditional Fouriertransform. The effect of the Fourier plane on positive-definiteness isclear: with α=1 the complex exponential is always either i, −i, 1, or −1since n is an integer and with the {a_(n)} all in the unit interval[0,1] there is no way to do anything with phase relationships. If α isnot an integer then nα in general is not, and thus the complexexponential can attain other values not on the unit circle.

Referring now to the arrangement of FIG. 3, the above relationship canbe used in sequence to calculate the phase contributions at each opticaltransfer function element 103 a, 103 b, etc. as well as any element inthe Fourier plane 104:

The distances between the elements can be used to calculate the power ofthe Fractional Fourier transform at the element location

-   -   These powers are used to determine the phase variation        introduced by that element    -   The composite effect of all optical transfer function elements        in the system can then be calculated. The phase shift introduced        by N optical transfer function elements for image expansion        coefficient f_(n,m) and optical transfer function element        expansion coefficients t¹ _(n,m) . . . t^(N) _(n,m); with        separation distances determining Fractional Fourier transform        powers α₁, . . . , α_(N) is:

$\begin{matrix}{\left( {\sum\limits_{j = 1}^{N}t_{n,m}^{j}} \right)^{\lbrack{{({{- }\; n\; {\pi/2}})}{\sum\limits_{j = 1}^{N}{\alpha \;}_{j}}}\rbrack}} & (9)\end{matrix}$

Approximation methods can then be used, based on a very small butsufficiently rich collection of easily implemented optical transferfunction elements and a convenient range of separation distancesdictated by physical considerations.

A class of system embodiments of special interest provided for by theinvention are those where a sequence or physically-adjacent stack ofprogrammable light modulators, such as LCD panels, are used as theoptical transfer function elements. Electrical and/or computer and/or,with future materials, optical control could then be used to alter theindividual transfer function of each optical transfer function element.The latter arrangement could be implemented as an integrated opticsdevice, perhaps including storage capabilities to storage the transferfunction pixel arrays and perhaps even an associated processor forcomputing the individual optical transfer function element transferfunction pixel arrays. The result is a powerful, relatively low-cost,potentially high-resolution real-time integrated optics image processorwhich would perform optical transformations otherwise requiring manyvast orders of magnitudes of computation power. Further, the imagesource and observation elements may in fact be optoelectric transducers(LED arrays, LCDs, CCDs, CRTS, etc.) and may be further integrated intoa comprehensive system which can further be abstracted into a moregeneral purpose complex-number arithmetic optical computing arrayprocessor of tremendous processing power. The invention provides for therange of these integrated optical processor architectures to readilyimplemented through photolithography or beam-controlled fabricationmethods.

Returning to the image processing application, such an integrated opticsimage processor could be realized for monochrome, visual color, orwide-spectrum images depending on the systems architecture and/or thecomponents used. For example, a visual color image processor could berealized by three monochrome image processing subsystems as describedabove, each monochrome image processing sub-systems dedicated to one ofthe three visual color components (i.e., “red,” “green,” and “blue”) asis commonly done in videographics and digital image processing systems.Alternatively, color LCD panels may be used as the optical transferfunction elements, allowing a single image processing system asdescribed in the previous paragraph to simultaneously process the entirevisual range. Both of these system and method approaches may also beused for wide-spectrum images: a wide-spectrum image may be decomposedinto multiple optical frequency bands which are processed separately(and potentially, though not necessarily, recombined), or wide-spectrumLCD optical transfer function elements can be used in a single channelsystem.

It is noted that the multichannel system has the additionalwide-spectrum advantage of permitting different materials to be used ineach spectral band, thus allowing choices of materials each best matchedfor its associated band. It is also noted that another wide-spectrumadvantage of the multichannel system is that splitting the spectrumpotentially allows for approximate wavelength-dependent dispersioncorrections and/or control. Yet another wide-spectrum advantage of themultichannel system is that splitting the spectrum potentially allowsfor separate detector areas far each wavelength band, having potentialapplications in astronomical observation sensors and wavelength-divisionmultiplexing. Finally, it is noted that both dispersioncompensation/control and separate detector partitions can be served witha large plurality of wavelength-partitioned channels readily implementedthrough photolithography or beam-controlled fabrication of theintegrated optical processor.

Before continuing with additional aspects of these several types ofintegrated apical processors, it: is noted that the systems and methodsdescribed herein can be applied to conventional fens-based optical imageprocessing systems as well as to systems with other types of elementsobeying Fractional Fourier optical models and as well to widely rangingenvironments such as integrated optics, optical computing systems,particle beam systems, radiation accelerators, and astronomicalobservation methods.

In particular it is noted that an important class of optical systemsobeying Fractional Fourier optical models are graded-index opticalmaterials (such as optical fibers), in particular quadratic graded-indexmedia. It is therefore noted that graded index materials can be used inthe fabrication of the integrated optical processors described above,removing the need for any lenses. This allows further design options inthe creation of both monolithic and hybrid physical realizations andfabrication of the integrated optical processors described above.

For instance, FIG. 4 shows the optics device of FIG. 3 configured as amonolithic device 120 providing optical filtering according to varioustechniques disclosed herein. One alternative is to omit image source 101and image observation element 106, resulting in optics device 125. Ifdesired, as shown in FIG. 5, optics device 100 may be implemented withinoptics system 130, thus defining an integrated optics system accordingto an implementation of the present invention.

It is next noted that the entire systems and methods developed thus farcan now be generalized away from the Fractional Fourier transform andits associated Herrnite function basis and quadratic phasecharacteristics. For example, the integrated optical processor can, asjust mentioned, be implemented with graded-index materials. To realizethe Fractional Fourier transform and its associated Hermite functionbasis and quadratic phase characteristics in the development so far,quadratic graded-index media must be used. If instead non-quadraticgraded-index media are used, a similar but different mathematicalframework can in general be employed: the Hermite functions will bereplaced with a different set of basis functions affiliated with thegraded-index profile, and the Fractional Fourier transform will bereplaced by a different operation affiliated with these replacementbasis functions. This approach can prove important and advantageous inmass production as specific types of non-quadratic graded-index mediamay be found to more readily lend themselves to the fabrication of thesame general form integrated optical processor architectures describedabove, as well as related ones.

Finally, returning now to the original Fractional Fourier transformapproach and framework, note that it is also possible to effectivelyturn the original purpose of the framework on its head and admitcontrollable filtering elements that are indeed non-positive-definite intheir transfer function, i.e., shifting the phase of the light. Suchcontrollable optical phase-shifting filter elements may be, for example,realized as two-dimensional arrays of twisting liquid crystals or otherelectronically controllable optical material which varies the phase oftransmitted light, for example, by changing its intrinsic opticalproperties such as its index of refraction, and/or changing theeffective propagation length of the transmitted light, or by yet othermethodologies.

One case of an optical image processing or computation system involvingthis technology would involve electronically controllable opticalphase-shifting filtering element arrays which can only vary thetransmitted light phase over some range, perhaps a small one. A largerphase variation is therefore possible if a plurality of such arrays areganged, i.e., either stacked together or distributed in space. Ingeneral this will result in some or all of the arrays to lie outside theFourier plane, particularly if the overall optical system incorporatingthis array plurality is physically compact. The mathematical frameworkdescribed earlier for calculating the overall transfer function assumingpositive-definite optical filter elements can be used directly: thephase perturbations made possible by each phase-controllable element areincorporated by simply substituting the real-valued array parameterswith complex-valued ones as is familiar in the common art of signalprocessing. The resulting mathematical expressions are then usable fortransfer function synthesis as described earlier, but now with theadditional degrees of freedom associated with the phase control madepossible by these elements.

A particularly useful application of this would be to use only all-passphase-variable elements in a compound optical system set up to operateas an electronically adjustable lens. By varying the phase-shifts of theall-pass filter elements, the Fractional Fourier transform effects of alens of varying focal length andor varying separation distance from theaffiliated optics can be synthesized. Various compound opticsarrangements can therefore be made utilizing this electronicallycontrolled synthesis of effective focal length and/or varying separationdistance. One important application would be the creation of anelectronically adjustable lens with variable focus and variable zoomcapabilities. The latter is a straightforward application of theelectronically controlled variable phase-shift aspect of the invention.

Conventional optics rely on varying physical separation distances ofmovable optical elements to realize zoom and focus effects, thus theelectronically controlled variable phase-shift realization of theinvention would simply replace one or more such moving optical elements.Particularly useful is an arrangement where the electronicallycontrollable optical phase-shifting filter elements involve no movingparts, thus allowing electronically controllable variable focus andvariable zoom capabilities to be realized with no moving parts. Not onlywill this replace the need for today's extensive precision optics andprecision mechanics, but the resulting optical effects can be made veryprecise and operate with fast dynamics. Because twisting liquid crystalstoday operate at full contrast at video rates (30 Hz), variations inzoom and focus would be executable in 30 msec or less. Further, with theaddition of electronic image sources and detectors on either side of thecontrollable optical phase-shifting filter element arrays, the moregeneral purpose complex-number optical computing array processor oftremendous processing power described earlier may also be fabricatedwith an optical architecture similar to the electronically variable lensjust described. Finally, it is noted that as material methods advance,the invention provides for the variable phase elements described aboveto be controlled optically in part or whole rather than exclusivelyelectrically.

All publications and patent applications mentioned in this specificationare herein incorporated by reference to the same extent as if eachindividual publication or patent application was specifically andindividually indicated to be incorporated by reference. The inventionnow being fully described, it will be apparent to one of ordinary skillin the art that many changes and modifications can be made theretowithout departing from its spirit or scope.

REFERENCES CITED

The following references are cited in this patent application:

-   [1] L. Levi, Applied Optics, Volume 2 (Section 19.2), Wiley, New    York, 1980.-   [2] J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, New    York, 1968.-   [3] K. lizuka, Engineering Optics, Second Edition, Springer-Verlag,    1987.-   [4] A. Papoulis, Systems and Transforms with Applications in Optics,    Krieger, Malabar, Fla., 1986.-   [5] L. F. Ludwig, “General Thin-Lens Action on Spatial Intensity    (Amplitude) Distribution Behaves as Non-Integer Powers of Fourier    Transform,” Spatial Light Madidators and Applications Conference,    South Lake Tahoe, 1988.-   [6] R. Dorsch, “Fractional Fourier Transformer of Variable Order    Based on a Modular Lens System,” in Applied Optics, vol. 34, no. 26,    pp. 6016-6020, September 1995.-   [7] E. U. Condon, “Immersion of the Fourier Transform in a    Continuous Group of Functional Transforms,” in Proceedings of the    National Academy of Science, vol. 23, pp. 158-161, 1937.-   [8] V. Bargmann, “On a Hilbert Space of Analytical Functions and an    Associated Integral Transform,” Comm. Pure Appl. Math, Volume 14,    1961, 187-214.-   [9] V. Namias, “The Fractional Order Fourier Transform and its    Application to Quantum Mechanics,” in J. of Institute of Mathematics    and Applications, vol. 25, pp. 241-265, 1980.-   [10] B. W. Dickinson and D. Steiglitz, “Eigenvectors and Functions    of the Discrete Fourier Transform,” in IEEE Transactions on    Acoustics, Speech, and Signal Processing, vol. ASSP-30, no. 1, Feb.    1982.-   [11] F. H. Kerr, “A Distributional Approach to Namias' Fractional    Fourier Transforms,” in Proceedings of the Royal Society of    Edinburgh, vol. 108A, pp. 133-143, 1983.-   [12] F. H. Kerr, “On Namias' Fractional Fourier Transforms,” in    IMA J. of Applied Mathematics, vol. 39, no. 2, pp. 159-175, 1987.-   [13] P. J. Davis, Interpolation and Approximation, Dover, N.Y.,    1975.-   [14] N. I. Achieser, Theory of Approximation, Dover, N.Y., 1992.-   [15] G. B. Folland, Harmonic Analysis in Phase Space, Princeton    University Press, Princeton, N.J., 1989.-   [16] N. N. Lebedev, Special Functions and their Applications, Dover,    N.Y., 1965.-   [17] N. Wiener, The Fourier Integral and Certain of Its    Applications, (Dover Publications, Inc., New York, 1958) originally    Cambridge University Press, Cambridge, England, 1933.-   [18] S. Thangavelu, Lectures on Hermite and Laguerre Expansions,    Princeton University Press, Princeton, N.J., 1993.-   [19] H. M. Ozaktas, D. Mendlovic, “Fractional Fourier Transforms and    their Optical Implementation II,” Journal of the Optical Society of    America, A 10, pp. 2522-2531, 1993.

1. An optical processing system, said system comprising: anelectronically-controllable image source for emitting an optical imageresponsive to at least one incoming electronic image signal; a firstimaging optical element having two distinct sides, the first imagingoptical element configured to accept the optical image at one of the twosides and to alter the optical image according to a imaging process, andto provide a first altered optical image from the second of the twosides of the element; a plurality of phase-shifting element arrays, atleast a plurality of which are responsive to associated incomingphase-shifting control signals associated with fractional Fourier planetransfer functions, and adjacently stacked so that consecutive pairs ofphase-shifting element arrays are immediately adjacent with one anotherso as to form a stack with a first image passing surface at one edge ofthe stack and a second image passing surface at the other edge of thestack, the plurality of phase-shifting element arrays arranged so thatthe first image passing surface receives the first altered opticalimage, the second image passing surface providing a second alteredoptical image; a second imaging image forming element arranged toreceive the second altered optical image at one of two sides; and anelectronic image sensor facing the second of the two sides of the secondimaging image forming element; wherein the emitted optical image isdirected through the first imaging image forming element, producing anoptical field directed through the stacked plurality of phase-shiftingelement arrays and subsequently directed through the second imagingimage forming element and sensed by the electronic image sensor.
 2. Theoptical processing system of claim 2 wherein the optical processingsystem is one from the group of: a monolithic integrated opticalprocessor and a hybrid integrated optical processor.
 3. The opticalprocessing system of claim 2 wherein the imaging material has aquadratic phase shift characteristic.
 4. The optical processing systemof claim 2 wherein at least one of the first imaging optical element andthe second imaging optical element comprises a lens.
 5. The opticalprocessing system of claim 2 wherein at least one of the first imagingoptical element and the second imaging optical element comprises GRINmaterial.
 6. The optical processing system of claim 2 wherein at leastone of the incoming phase-shifting control signals comprises informationcalculated by an associated processor.
 7. The optical processing systemof claim 2 wherein at least one of the incoming phase-shifting controlsignals comprises information stored in an associated memory.
 8. Theoptical processing system of claim wherein at least one of the pluralityof phase-shifting element arrays is implemented with an liquid crystaldisplay.
 9. The optical processing system of claim 2 wherein at leastone of the plurality of phase-shifting element arrays shifts the phaseof light transmitted through it.
 10. An optical processing system, saidsystem comprising: an electronically-controllable image source foremitting an optical image responsive to at least one incoming electronicimage signal to create an emitted optical image; a first optical elementhaving two distinct sides, accepting the emitted optical image at one ofthe two sides of the element, altering the emitted optical image, andproviding a first altered light from the second of the two sides; astacked plurality of phase-shifting element arrays, at least a pluralityof which are responsive to associated incoming phase-shifting controlsignals, the resulting stack providing a second altered light, andwherein incoming control signal is associated with transfer functionsassociated with fractional Fourier planes; a second optical elementarranged to receive the second altered light; and an electronic imagesensor facing the opposite side of the second optical element, theelectronic image sensor operating as an optical observation element;wherein the emitted optical image is directed through the first opticalelement, producing an optical field directed through the stackedplurality of phase-shifting element arrays and subsequently directedthrough the second optical element and sensed by the electronic imagesensor.
 11. The optical processing system of claim 12 wherein at leastone of the electronically-controllable image source and the electronicimage sensor comprises an LED array.
 12. The optical processing systemof claim 12 wherein the optical processing system is configured as ahybrid physical realization of an integrated optical processor.
 13. Theoptical processing system of claim 12 wherein at least one of theincoming phase-shifting control signals comprises information calculatedby an associated processor.
 14. The optical processing system of claim12 wherein at least one of the incoming phase-shifting control signalscomprises information stored in an associated memory.